In this section, we use matrices to give a representation of complex numbers. Indeed, consider the set
We will write
Clearly, the set is not empty. For example, we have
In particular, we have
for any real numbers a, b, c, and d.
Algebraic Properties of
- Addition: For any real numbers a, b, c, and d, we have
Ma,b + Mc,d = Ma+c,b+d.
In other words, if we add two elements of the set , we still get a matrix in . In particular, we have
-Ma,b = M-a,-b.
- Multiplication by a number: We have
So a multiplication of an element of and a number gives a matrix in .
- Multiplication: For any real numbers a, b, c, and d, we have
In other words, we have
Ma,b Mc,d = Mac-bd, ad+bc.
This is an extraordinary formula. It is quite conceivable given the difficult form of the matrix multiplication that, a priori, the product of two elements of may not be in again. But, in this case, it turns out to be true.
The above properties infer to a very nice structure. The next natural question to ask, in this case, is whether a nonzero element of is invertible. Indeed, for any real numbers a and b, we have
So, if , the matrix Ma,b is invertible and
In other words, any nonzero element Ma,b of is invertible and its inverse is still in since
In order to define the division in , we will use the inverse. Indeed, recall that
So for the set , we have
Ma, b÷Mc, d = Ma, b×Mc, d-1 = Ma, b×M,
Ma, b÷Mc, d = M, -
The matrix Ma,-b is called the conjugate of Ma,b. Note that the conjugate of the conjugate of Ma,b is Ma,b itself.
Fundamental Equation. For any Ma,b in , we have
Ma,b = a M1,0 + b M0,1 = a I2 + b M0,1.
M0,1 M0,1 = M-1,0 = - I2.
Remark. If we introduce an imaginary number i such that i2 = -1, then the matrix Ma,b may be rewritten by
a + bi